M 595 – Two Solutions
From Congruence Exercises
5.
One of these numbers is not congruent to the others: 30, 69, 97, 134, 173
Which of the numbers is not congruent to the others and what is the modulus?
Consider all the differences and factor them:
69 – 30 = 39
39 = 3

13
97 – 30 = 67
67 is prime
97 – 69 = 28
28 = 2

2

7
134 – 30 = 104 173 – 30 = 143
104 = 2

2

2

13 143 = 11

13
134 – 69 = 65 173 – 69 = 104
65 = 5

13 104 = 2

2

2

13
134 – 97 = 37 173 – 97 = 76
37 is prime 76 = 2

2

19
173 – 134 = 39
39 = 3

13
None of the differences involving 97 have factors in common.
All of the others have a factor of 13, so the other four are all

(mod 13).
From Solving Congruences
2.
Find the smallest natural number n , where 5 n

3 (mod 7) and 2 n

1 (mod 5).
[careful: those moduli are different]
If 5 n

3 (mod 7), then n

3(5 n )

3(3) = 9

2 (mod 7)
If 2 n

1 (mod 5), then n

3(2 n )

3(1) = 3 (mod 5)
So we need a number n that is

2 (mod 7) and is

3 (mod 5)
These are all

2 (mod 7): 2, 9, 16, 23, 30, 37, 44, 51,…
These are all

3 (mod 5): 3, 8, 13, 18, 23, 28, 33, 38,…
The smallest number in both is 23.